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208 CHAPTER 5 / FUNCTION MINIMIZATION

(uncomplemented /•) R-M expansion (PPRME). Each R-M coefficient is the set

(5-18)

m
obtained from the subnumbers of / by replacing m 1 's with O's in 2 possible ways in the
binary number corresponding to decimal i :

go = fo •••000
gi = e/(i, o) = /, e f 0 ... 001 ... ooo
= e/(2, 0) = h e /o ... 010 ... ooo
§2
£ = e/(3, 2, i, o) = / e / e /i e /o • • • 01 1 • • • 010 • • • 001 • • • ooo
3
3
2
84 = 0/(4, 0) = / 4 © /o ... 100 ... 000
= e/(5, 4, i, o) = / e / e /i e / • • • 101 • • • 100 • • • 001 • • • ooo
g5 5 4 0

= ©/•

Note that any g, in Eq. (5.18) is 1 if an odd number of / coefficients are logic 1, but is
0 if an even number of / coefficients are logic 1. If a Karnaugh map (K-map) of F n is
available, the values for the g, are easily determined by counting the 1 's in the map domains
defined by the O's in the binary number representing / in g/. For example, the value of #5
is found by counting the 1's present in the *o*2 domain for a function FA, = (*o*i*2*3).
Thus, gs = 1 if an odd number of 1 's exists or g 5 = 0 otherwise. Similarly, to determine
the logic value for g§ one would count the number of 1's present in the x 1*2*3 domain for
the same function, etc. All terms in the PPRME expansion whose g coefficients are logic 0
are disregarded.

5.6 THE POS-TO-EQPOS REED-MULLER TRANSFORMATION

The dual of Eqs. (5.16) is the generalization of Corollary II (Subsection 3.11.1) and is
expressed as

2"-l
F n(xo, x\, *2,..., *H-I) = 1 \(Mi + fi)
i=0

i=0
= (M 0 + /o) O (Mi + /i) O (M 2 + / 2)
0---0(M 2 -_,+/2.-i), (5.19)